In the formulas below, a multiplication between a matrix and a pair of
coordinates should be carried out regarding the pair as a column
vector (or a matrix with two rows and one column). Thus
(x,y)=(ax+by, cx+dy).

Translation by (x,y):

(x,y)(x+x, y+y)

Rotation through (counterclockwise) around the origin:

Rotation through (counterclockwise)
around an arbitrary point (x,y):

Reflection in the x-axis:

(x,y)(x,-y)

Reflection in the y-axis:

(x,y)(-x,y)

Reflection in the xy-diagonal:

(x,y)(y,x)

Reflection in a line with equation ax+by+c=0:

Reflection in a line going through (x,y) and making an
angle with the x-axis:

Glide-reflection in a line L with displacement d: Apply first
a reflection in L, then a translation by a vector of length d in
the direction of L, that is, by the vector